A100851 Triangle read by rows: T(n,k) = 2^n * 3^k, 0 <= k <= n, n >= 0.

1, 2, 6, 4, 12, 36, 8, 24, 72, 216, 16, 48, 144, 432, 1296, 32, 96, 288, 864, 2592, 7776, 64, 192, 576, 1728, 5184, 15552, 46656, 128, 384, 1152, 3456, 10368, 31104, 93312, 279936, 256, 768, 2304, 6912, 20736, 62208, 186624, 559872, 1679616, 512, 1536, 4608, 13824, 41472, 124416, 373248, 1119744, 3359232, 10077696

Offset: 0

Author

Reinhard Zumkeller, Nov 20 2004

Keywords

• nonn
• tabl
• easy

Examples

			From _Stefano Spezia_, Apr 28 2024: (Start)
Triangle begins:
   1;
   2,  6;
   4, 12,  36;
   8, 24,  72, 216;
  16, 48, 144, 432, 1296;
  32, 96, 288, 864, 2592, 7776;
  ...
(End)
		

Links

• G. C. Greubel, Rows n = 0..100 of the triangle, flattened
• Eric Weisstein's World of Mathematics, Smooth Number.

Crossrefs

Cf. A000079, A000400, A001021, A003586, A005010, A007283, A016129.

Cf. A016149, A051958, A053524, A100852, A248337.

Cf. A065333(T(n, k))=1.

Programs

Magma

A100851:= func< n,k | 2^n*3^k >;
[A100851(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 11 2024

Mathematica

A100851[n_, k_]= 2^n*3^k;
Table[A100851[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 11 2024 *)

SageMath

def A100851(n,k): return 2^n*3^k
flatten([[A100851(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 11 2024

Formula

T(n,0) = A000079(n).

T(n,1) = A007283(n) for n>0.

T(n,2) = A005010(n) for n>1.

T(n,n) = A000400(n) = A100852(n,n).

Sum_{k=0..n} T(n, k) = A016129(n).

T(2*n, n) = A001021(n). - Reinhard Zumkeller, Mar 04 2006

G.f.: 1/((1 - 2*x)*(1 - 6*x*y)). - Stefano Spezia, Apr 28 2024

From G. C. Greubel, Nov 11 2024: (Start)

Sum_{k=0..n} (-1)^k*T(n, k) = A053524(n+1).

Sum_{k=0..floor(n/2)} T(n-k, k) = (1/2)*((1-(-1)^n)*A248337((n+1)/2) + (1 + (-1)^n)*A016149(n/2)).

Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (1/2)*(-1)^floor(n/2)*( (1+(-1)^n) *A051958((n+2)/2) + 2*(1-(-1)^n)*A051958((n+1)/2)). (End)

Sum_{n>=0, k=0..n} 1/T(n,k) = 12/5. - Amiram Eldar, May 12 2025